Question

nth term of a Sequence Find the nth term of a sequence whose first several terms are given.

$5,-25,125,-625,$

Short answer

The nth term of the sequence is$a_n={\left(-1\right)}^{n-1}{5}^n$.

Step-by-step solution

Step 1.  Given

The given sequence is 5, -25, 125, -625,…

Step 2.  To determine

We have to find thenth term of the sequence.

Step 3.  Calculation

The given sequence 5, -25, 125, -625…. is a geometric sequence with common ratio

$d=\frac{\text{2nd} \text{term}}{1\text{st} \text{term}}=\frac{-25}{5}=-5$.

First term is $a_1=5$.

Using the term formula for geometric sequence we get:

$\begin{array}{c}{a}_n=a_1{r}^{n-1}\\ =5{\left(-5\right)}^{n-1}\\ =5{\left(-1\times 5\right)}^{n-1}\\ =5\times {\left(-1\right)}^{n-1}\times {\left(5\right)}^{n-1}\\ ={\left(-1\right)}^{n-1}{5}^n\end{array}$

Hence, the nth term of the sequence is $a_n={\left(-1\right)}^{n-1}{5}^n$.